3.4.51 \(\int e^x \text {sech}(e^x) \, dx\) [351]

Optimal. Leaf size=5 \[ \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \]

[Out]

arctan(sinh(exp(x)))

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Rubi [A]
time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 3855} \begin {gather*} \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*Sech[E^x],x]

[Out]

ArcTan[Sinh[E^x]]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int e^x \text {sech}\left (e^x\right ) \, dx &=\text {Subst}\left (\int \text {sech}(x) \, dx,x,e^x\right )\\ &=\tan ^{-1}\left (\sinh \left (e^x\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 5, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*Sech[E^x],x]

[Out]

ArcTan[Sinh[E^x]]

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(11\) vs. \(2(5)=10\).
time = 2.06, size = 9, normalized size = 1.80 \begin {gather*} 2 \text {ArcTan}\left [\text {Tanh}\left [\frac {E^x}{2}\right ]\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[E^x*Sech[E^x],x]')

[Out]

2 ArcTan[Tanh[E ^ x / 2]]

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Maple [A]
time = 0.01, size = 5, normalized size = 1.00

method	result	size
derivativedivides	\(\arctan \left (\sinh \left ({\mathrm e}^{x}\right )\right )\)	\(5\)
default	\(\arctan \left (\sinh \left ({\mathrm e}^{x}\right )\right )\)	\(5\)
risch	\(i \ln \left ({\mathrm e}^{{\mathrm e}^{x}}+i\right )-i \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-i\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*sech(exp(x)),x,method=_RETURNVERBOSE)

[Out]

arctan(sinh(exp(x)))

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Maxima [A]
time = 0.27, size = 4, normalized size = 0.80 \begin {gather*} \arctan \left (\sinh \left (e^{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sech(exp(x)),x, algorithm="maxima")

[Out]

arctan(sinh(e^x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (4) = 8\).
time = 0.38, size = 16, normalized size = 3.20 \begin {gather*} 2 \, \arctan \left (\cosh \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sinh \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sech(exp(x)),x, algorithm="fricas")

[Out]

2*arctan(cosh(cosh(x) + sinh(x)) + sinh(cosh(x) + sinh(x)))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \operatorname {sech}{\left (e^{x} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sech(exp(x)),x)

[Out]

Integral(exp(x)*sech(exp(x)), x)

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Giac [A]
time = 0.00, size = 6, normalized size = 1.20 \begin {gather*} 2 \arctan \left (\mathrm {e}^{\mathrm {e}^{x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sech(exp(x)),x)

[Out]

2*arctan(e^(e^x))

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Mupad [B]
time = 0.05, size = 6, normalized size = 1.20 \begin {gather*} 2\,\mathrm {atan}\left ({\mathrm {e}}^{{\mathrm {e}}^x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/cosh(exp(x)),x)

[Out]

2*atan(exp(exp(x)))

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